### Topic #3: Cooperative Binding Calculations

Model reaction:     PA2 <=> P + 2 A
Dissociation constant: Kd = [P][A]nH/[PA2]   (and nH < 2, for this example)
Fractional occupancy: Y = [A]nH/(KdnH + [A]nH)

[Note: P and A are used on this page (instead of the more general, M and L) to indicate a particular binding reaction under study. Any symbols can be (and are) used to represent the macromolecule and ligand, respectively.]

The Hill equation is derived from the expressions for Kd and Y. In logarithmic form:
log(Y/1-Y) = nHlogA - nHlogKd ,  where
nH = the Hill coefficient (maximum slope of the Hill plot);
A = [A]free;
Kd = the dissociation constant.
We have a 0.4 mg/ml solution of a protein (Mr = 40,000 Da). The stoichiometry of A binding is 2.0. When bound, the UV absorption of the ligand changes. Thus, we can determine [A]bound directly. We want to determine Kd and the Hill coefficient, nH. This is done in three steps.
1. Enter a value for [A]total in this box:
 [A] added = mM 2. For the above value of [A]total, Calculate [A]bound [A]bound = mM
(Each calculated [A]bound value has a small "experimental error" added to it.
Tabbing out of the volume entry slot or clicking anywhere on the page will also calculate [L]free.)

3. Record the values of [A]bound you obtain. Then, calculate [A]free, Y, and (Y/1-Y). Finally, graph the values on a Hill plot to determine Kd and nH. You should get enough data so as to have 3 or 4 values of Y, both above and below the Kd value.
(Hint: Graph the values you obtain as they are calculated; then as the shape of the curve becomes apparent, choose values for [A]total that fall into the appropriate range.)